Express $(4^3)^{\frac{1}{5}}$ in simplest radical form.
Understand the Problem
The question asks to express $(4^3)^{\frac{1}{5}}$ in its simplest radical form. First simplify $4^3$, and then rewrite the results as a radical.
Answer
$2\sqrt[5]{2}$
Answer for screen readers
$2\sqrt[5]{2}$
Steps to Solve
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Simplify the expression inside the parenthesis $4^3$ means 4 multiplied by itself three times: $ 4^3 = 4 \cdot 4 \cdot 4 = 64 $ So we have $ (4^3)^{\frac{1}{5}} = 64^{\frac{1}{5}} $.
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Convert the fractional exponent to radical form A fractional exponent $\frac{1}{n}$ is equivalent to taking the $n$-th root. Therefore, $64^{\frac{1}{5}}$ can be written as $\sqrt[5]{64}$.
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Simplify the radical We want to find the simplest radical form of $\sqrt[5]{64}$. Since $64 = 2^6$, we have: $ \sqrt[5]{64} = \sqrt[5]{2^6} $ We can rewrite $2^6$ as $2^5 \cdot 2$: $ \sqrt[5]{2^6} = \sqrt[5]{2^5 \cdot 2} $ Now, we can take out the $2^5$ from under the radical: $ \sqrt[5]{2^5 \cdot 2} = 2\sqrt[5]{2} $
$2\sqrt[5]{2}$
More Information
The fifth root of 64 can be expressed as 2 times the fifth root of 2.
Tips
A common mistake might be not simplifying the radical completely. For example, expressing $64^{\frac{1}{5}}$ as $\sqrt[5]{64}$ might be considered incomplete if the instructions ask for the simplest radical form.
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